Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q25

Expert-verified

Geometry

Found in: Page 176

Book edition Student Edition

Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Pages 227 pages

ISBN 9780395977279

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

Write a paragraph proof.
Given: $▱ A B C D; ▱ B E D F$
Prove: AECF is a $▱$ .
(Hint: A short proof is possible if certain auxiliary segments ate drawn.)

Two pairs of opposite sides $\vec{A E} ≅ \vec{F C}$ and $\vec{A F} ≅ \vec{C E}$ in AECF are parallel therefore, AECF is a parallelogram.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Relation between triangle DAE and triangle BCF

In triangles $Δ D A E$ and $Δ B C F$ .

$\vec{A D} ≅ \vec{B C} ∴$ (ABCD is a rhombus)

$∠ A D C ≅ ∠ A B C ∴$ (ABCD is a rhombus)

$∠ A D E ≅ ∠ C B F ∴$ ( $∠ E D C & ∠ A B F$ are the equal angle of parallelogram made of parallel sides two parallelograms)

ABCD is a parallelogram.

Similarly, $∠ A D C ≅ ∠ A B C$ ( ABCD is a rhombus),

$∠ E D C & ∠ A B F$ are equal angles of parallelogram made of parallel sides.

$∠ A D E ≅ ∠ C B F$

$\vec{D E} ≅ \vec{B F}$ $(∴ B E D F is a parallelogram .)$

Therefore, by SAS postulate $Δ D A E ≅ Δ B C F$ .

By CPCT, $\vec{A E} ≅ \vec{F C}$ .

Step 2. Relation between triangle AFB and triangle CDE

In triangles $Δ A F B & Δ C D E$ .

$\vec{D E} ≅ \vec{B F} ∴ (B E D F is a parallelogram)$

Since, $∠ E D C & ∠ A B F$ are equal angles of parallelogram made of parallel sides.

$∠ A D E ≅ ∠ C B F$

$\vec{D C} ≅ \vec{B A} ∴ (B E D F is a parallleogram)$

Therefore, by SAS postulate $Δ A F B ≅ Δ C E D$ .

By CPCT, $\vec{A F} ≅ \vec{C E}$ .

Step 3. State the conclusion

Since two pairs of opposite sides $\vec{A E} ≅ \vec{F C}$ and $\vec{A F} ≅ \vec{C E}$ in AECF are parallel, therefore, AECF is a parallelogram.

Most popular questions for Math Textbooks

PQRS is a parallelogram. Find the values of a, b, x and y.

For exercises, 14-18 write paragraph proofs.

Given: parallelogram ABCD, $\bar{AN}$ bisects $∠ DAB$ ; $\bar{CM}$ bisects $∠ BCD$ .

Prove: AMCN is a parallelogram.

Find the perimeter of $▱ RISK$ , if $RI = 17$ and $IS = 13$ .

Must quad. EFGH be a parallelogram? Can it be a parallelogram? Explain.

Must quad. EFGH be a parallelogram? Can it be a parallelogram? Explain.

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free